1) Represent repeated multiplication with exponents.

Any exponent can be represented with repeating multiplication by simply multiplying the base with the base by the number of the exponent.

Ex. 4^3 = 4\cdot4\cdot4

2) Describe how powers represent repeated multiplication.

Everyone knows how to calculate the expression 5\cdot5 (which is repeated addition), you can rewrite this in a shorter way by using powers. 5\cdot5 = 5^2. The number 5 is the base and number 2 is the exponent, the exponent is there to tell you how many times you use the base as a factor.

3) Demonstrate the difference between the exponent and the base by building models of a given power, such as 2^3 and 3^2.

                                                   

2^3 (the one on the left) is wanting to find the volume, length x width x height. 3^2 is finding the area or the shape, length x width.

4) Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2^3 and 3^2.

2^3 = 2 x 2 x 2 = 8

3^2 = 3 x 3 = 9

Although these powers look similar they are actually very different. As i stated before in question 1, you multiply the base with the base by the number the exponent is.

5) Evaluate powers with integral bases (excluding base 0) and whole number exponents.

Some examples for exponents and a review of the explanation written above i will show you some exponents and how to evaluate them.

4^3 = 4 x 4 x 4 = 64

1^7 = 1 x 1 x 1 x 1 x 1 x 1 x 1 = 1

10^3 = 10 x 10 x 10 = 1000

5^2 = 5 x 5 = 25

6) Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)^4, (-2^4), and -2^4

Now that you have a good understanding of plain exponents, we can add in parentheses and negative signs.

Ex 1. (-2)^4 = (-2)(-2)(-2)(-2) = 16

Ex 2. (-2^4) = (-1)(2)(2)(2)(2) = -16

Ex 3. -2^4 = (-1)(2)(2)(2)(2) = -16

A common mistake people make are that they don’t pay attention to where the parentheses are located in the equation. As shown in Ex 1 = 16 because the base is -2 and an even amount of negatives make a positive. Where Ex 2 and Ex 3 have 2 as the base with -1 as the coefficient before it.

7) Explain the exponent laws for multiplying and dividing powers with the same base.

The product law tells us that we need to keep the base, add the exponents together, and multiply the coefficient. The quotient law tells us that we need to keep the base, subtract the exponents, and divide the coefficients.

Multiplication: 4^5\cdot4^3 = 4^{5+3} = 4^8  Division: 4^5\div4^3 = 4^{5-3} = 4^2

8) Explain the exponent laws for raising a product and quotient to an exponent.

When you raise a product and quotient to an exponent which is known as the power law, you need to 1. keep the base 2. multiply the exponents. Remember that the outside exponent affects everything inside that brackets, do not forget the coefficient.

Ex 1 (3^2)^2 = 3^{2\cdot2} = 3^4

Ex 2. (5\cdot3^2)^2 = (5\cdot5)(3^{2\cdot2}) = 25\cdot3^4

9) Explain the law for powers with an exponent of zero.

Any power that has an exponent of zero, will equal 1.

n^0 = 1, n ≠ 0

10) Use patterns to show that a power with an exponent of zero is equal to one.

When you use the quotient law to evaluate 3^4\div3^4 you get 3^{4-4} = 3^0. When we use BEDMAS to solve 3^4\div3^4 we get 81\div81 = 1. Since 3^0 and 1 are the answers to the same question, 3^0 = 1.

11) Explain the law for powers with negative exponents.

Any base (except 0) raised to a negative exponent equals:

1. Reciprocal base

2. Make exponent positive

Example: *Coding is messed up, look under the expression for the example written in words*

$latex \frac({3}{5})^-2$ = \frac({5}{3})^2 = \frac{25}{9}

*3 over 5 to the power of negative 2 = 5 over 3 to the power of 2 = 25 over 9*

12) Use patterns to explain the negative exponent law.

Example: If your base was 2 you would divide by 2 each time.

2^5 = 32

2^4 = 16

2^3 = 8

2^2 = 4

2^1 = 2

2^0 = 1

2^{-1} = \frac{1}{2}

2^{-2} = \frac{1}{4}

2^{-3} = \frac{1}{8}

13) I can apply the exponent laws to powers with both integral and variable bases.

Yes you can! The laws used to solve integral exponents have the exact same formula for bases that are variables.

Examples:

Product Law- 2^4\cdot 2^2 = 2^{4+2} = 2^6

Product Law (Variable)- n^4\cdot n^2 = n^{4+2} = n^6

Quotient Law- 2^4\div 2^2 = 2^{4-2} = 2^2

Quotient Law (Variable)- n^4\div n^2 = n^{4-2} = n^2

Power Law- (2^4)^2 = 2^{4\cdot 2} = 2^8

Power Law (Variable)- (n^4)^2 = n^{4\cdot 2} = n^8

14) I can identify the error in a simplification of an expression involving powers.

I know how to find errors when simplifying expressions that involve powers because i have practiced and have learned each law. Its crucial that you understand each law so you don’t mess up by doing too many steps or doing too little amount of steps.

Ex 1. 4^2\cdot4^3 = 16^5 The mistake in this expression is that the person multiplied the bases together. When doing the product law you only multiply the coefficients. Correct answer: 4^2\cdot4^3 = 4^5

Ex 2. (n^{-3})^{-2} = n^{-5} The mistake in this expression is that the person added the exponents. When using the power law you must multiply the exponents instead of adding like the product law.  Correct answer: (n^{-3})^{-2} = n^{6}

15) Use the order of operations on expressions with powers.

You use order of operations on expressions with powers when the bases are nonidentical, when you are adding or subtracting exponents, or when none of the exponent laws can apply to the problem.

Ex. 2^4\cdot3^2 = 16\cdot9 = 144

Ex. 3^3\cdot2^1 = 27\cdot2 = 54

16) Determine the sum and difference of two powers.

When you are finding the sum and difference of two powers, all you need to do is BEDMAS

Ex. 4^2-6^2 = 16-36 = -20

Ex. 5^1+3^4 = 5+81 = 86

17) Identify the error in applying the order of operations in an incorrect solution.

You can tell if people don’t follow the order of operation (BEDMAS) very easily. Most people just do the equation from left to right.

Example of the incorrect way to do order of operation (BEDMAS):

2^4-8\div2+(-3)^2 = 2^4-8\div2+9 = 16-8\div2+9 = 8\div2+9 = 4+9 = 13

Instead of following the order of operation they did a step incorrectly in the yellow highlighted area.

This is how the correct answer would be shown

2^4-8\div2+(-3)^2 = 2^4-8\div2+9 = 16-8\div2+9 = 16-4+9 = 12+9 = 21

18) Use powers to solve problems (measurement problems).

In order to find the area of the shaded part in this square you must find the area of the larger square and then find the area of the smaller square that is inside the larger square, one you have found the areas you then subtract them to find the area of the shaded region. Do not forget the unit. 

15^2-5^2 = 225-25 = 200mm^2

19) Use powers to solve problems (growth problems).

A patch of grass doubles every 2 hours. Currently there are 10 leaves of grass. How many leaves of grass would be produced over the following times.

a) 2 hour

10(2^1) = 10(2) = 20 leaves of grass

b) 4 hours

10(2^2) = 10(4) = 40 leaves of grass

c) 6 hours

10(2^3) = 10(8) = 80 leaves of grass

d) n hours

10(2^n\frac2) *10(2 to the power of n over 2)*

When we use powers to solve growth problems we need to know how much is growing at an amount of time and how much you have before the growing. In my example you start with 10 leaves of grass and you double the amount you have every 2 hours. In part a, 2 hours was given for the patch of grass. Since in 2 hours the patch should double in size you just needed to double the amount you had before which is 10 leaves of grass. You do the same process for par b and c, but in part d there is a variable. Since there is a variable you don’t know how many hours you are given to let the patch grow which means you cant evaluate it, so all you can do is write the formula.

20) Applying the order of operations on expressions with powers involving negative exponents and variable bases.

In this expression there are steps you need to do to get the correct answer. What i did was i got rid of all the negative exponents and turned them positive and also reduced the coefficients in the first step, after that i subtracted the exponents like the quotient law tells us to. Once i finished subtracting them i wrote the answer in the final form. Other equations will have other procedures. Some will use the other laws, it is important to remember all the exponent laws.

A couple of extra things i know about exponents:

  • If the base is one the answer will always be 1
  • Coefficients show repeated addition
  • If you have a negative base and your exponent has an even amount of negatives it will be positive but if you have an odd number of negative bases it will be negative

3 thoughts on “Everything I Know About Exponents”

  1. The models for 3 is accurate and explained well and can be understand easily. Examples are well used and states the reasons and explanations lots. Explained a lot of details such as reasoning and errors. Used mathematical terms such as exponent laws well. Have lots of examples for 13. Zero exponent law is accurately explained. More examples with multiplying or dividing two powers with coefficients could be added. Should have examples for adding and subtracting powers with order of operation for 15. Could correct the wrong example for 17. Maybe add an example of using BEDMAS for multiplying or dividing powers and correct it.No photo to be seen on 20. but explained well for how to get the answer for the equation.

    Reply

    1. Good Job Henry, although I would have like to have seen you make the changes suggested by Wendy. See your Onenote for specific feedback.

      Reply

  2. Henry’s post on exponents is very detailed and is very well written with good examples and it was easy to understand what he was trying to show. -Henry’s Father

    Reply

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